Sports cars generally put driver involvement and road response near the center of the design. Luxury performance cars combine strong performance with a broader expectation of refinement, comfort, and convenient travel. The difference is the balance of priorities, not a rule that one is faster or the other cannot handle well.
The terms overlap. A sports car can have an expensive, comfortable cabin, and a luxury performance car can have excellent steering, brakes, and circuit capability. Look at how the actual vehicle allocates space, weight, equipment, and control settings. Those choices explain more than its price or the badge on the trunk.